User blog:B1mb0w/Recursion and Summation Functions
'Recursion Notation & Summation Function' I use short hand notation for recursion of nested functions. They are a further development of notation I use that is not in common use. 'Review of my Basic Notation' Here is a summary review of my notation: Decremented Function \(C\) is short-hand notation. For any arbitrary function: \(M^{c + 1}(n)\) then \(C = M^{c}(n)\) e.g. \(M^{c + 1}(n) = M© = M(M^c(n))\) Parameter Subscript Brackets is shorthand for functions with multiple parameters: \(M(x,0_{2}) = M(x,0,0)\) \(M(x,y_{2}) = M(x,y_1,y_2)\) \(M(x,0_{2},y_{3},z) = M(x,0,0,y_1,y_2,y_3,z)\) Leading Zeros Assumption applies to any function I define which accepts a variable number of input parameters. All leading zeroes can then be ignored: \(M(0_{x},0_{2},y_{3},1) = M(0_{+ 2},y_{3},1) = t_0(y_{3},1) = M(y_1,y_2,y_3,1)\) This assumption does not apply to functions I define which require a fixed number of input parameters. 'Recursion Notation' I use the following notation to standardise recursion of nested functions. Recursion Parameter Subscript \(*\) is used to define how recursion applies to multi-parameter functions, e.g.: \(M^{c + 1}(1,n_*) = M(1,C) = M(1,M^c(1,n_*))\) \(M^{c + 1}(1_*,n) = M(C,n) = M(M^c(1,n_*),n)\) Nested Recursion the recursion parameter subscript can be used more than once in nested functions, e.g.: \(W^{2}(x_*,W^{3}(x,y_*,z),z)\) \(= W^{2}(x_*,W(x,W^{2}(x,y_*,z),z),z)\) \(= W^{2}(x_*,W(x,W(x,W(x,y,z),z),z),z)\) \(= W(W(x,W(x,W(x,W(x,y,z),z),z),z),W(x,W(x,W(x,y,z),z),z),z)\) Indexed Recursion is useful short hand notation (index) for a function within a nested stack, e.g.: \(K^{5}(a,b_*) = K(a,K(a,K(a,K(a,K(a,b)))))\) then indexing notation is useful shorthand for these nested functions. \(K^{5}(a,L(K^{4}(a,b_*))_*) = K(a,L(K(a,K(a,K(a,K(a,b)))))) = K(a,L(K^{4}(a,b_*)))\) and \(K^{5}(a,L^{2}(K^{2}(a,b_*))_*) = K(a,K(a,K(a,L^{2}(K(a,K(a,b)))))) = K^{3}(a,L^{2}(K^{2}(a,b_*))_*)\) The left hand notations are more useful then the more standard notations in the middle or right hand because: * obviously when the exponent (e.g. 5) becomes very large, the middle notation becomes difficult or impossible to expanded using finite resources * the right hand notation appears simpler than the left hand but when the exponents (e.g. 1 and 4 or 3 and 2) become very large, then evaluating the size of the resulting function requires addition of very large numbers (in this case 1 + 4 = 5 or 3 + 2 = 5). The left hand notation avoids the need to add any numbers by simply stating the size of the resulting function exponent (e.g. 5). * the left hand notation is used to define my Quantum Function when the exponents can by very large googological numbers. Indexing multiple nested functions examples: \(K^{5}(a,L(K^{4}(a,L^{2}(K^{2}(a,b_*))_*))_*)\) \(= K(a,L(K^{4}(a,L^{2}(K^{2}(a,b_*))_*)))\) \(= K(a,L(K(a,K^{3}(a,L^{2}(K^{2}(a,b_*))_*))))\) \(= K(a,L(K(a,K(a,L^{2}(K^{2}(a,b_*))))))\) \(= K(a,L(K(a,K(a,L^{2}(K(a,K(a,b)))))))\) or \(K^{5}(a,L(K^{4}(a,L^{2}(K^{2}(a,b_*))_*))_*) = K(a,L(K^{2}(a,L^{2}(K^{2}(a,b_*))_*)))\) Here are some rules when using Index Recursion notation: Rule i1: \(K^{c}(a,L(K^{d}(a,b_*))_*)\) only when \(0 < d < c\) Rule i2: \(K^{c}(a,L(K^{d}(a,L^{f}(K^{e}(a,b_*))_*))_*)\) only when \(0 < e < d < c\) Rule i3: \(K^{c + 1}(a,L(K^{d}(a,L^{f}(K^{e}(a,b_*))_*))_*)\) \(= K(a,C)\) when \(0 < e < d < c\) \(= K(a,L(K^{d}(a,L^{f}(K^{e}(a,b_*))_*)))\) when \(0 < e < d = c\) Rule i4: \(K^{c}(a,K(a,(K^{d}(a,b_*))_*) = K^{c + 1}(a,(K^{d}(a,b_*)_*)\) for any \(0 < d < c\) Indexed Recursion (part two) includes short hand notation (index) for function parameters that appear after the recursion parameter subscript has been applied, e.g.: \(V^{3}(1,2_*) = V(1,V(1,V(1,2)))\) requires no further notation \(V^{3}(1_*,2) = V(V(V(1,2),2),2)\) implies additional index notation \(V^{3}(1_*,2) = V^{3}(1_*,22)\) where \(22 = 2\) this allows specifying changing values of the latter parameters within nested functions. Here are some examples: \(V^{3}(1_*,20) = V(V(V(1,2),2),0)\) \(V^{3}(V^{2}(1_*,21)_*,20) = V(V(V(1,2),1),0)\) \(V^{3}(V^{2}(1_*,21)_*,22) = V(V(V(1,2),1),2)\) and \(22 = 2\) \(V^{5}(V^{3}(1_*,21)_*,20) = V(V(V(V(V(1,2),2),1),2),0)\) Indexed Recursion (part three) elaborates this notation for the following edge case: \(V^{3}(1_*,2) = V(V(V(1,2),2),2)\) implies index notation \(V^{3}(1_*,2) = V^{3}(1_*,22)\) \(V^{3}(1_*,3) = V(V(V(1,3),3),3)\) implies index notation \(V^{3}(1_*,3) = V^{3}(1_*,33)\) where \(V^{3}(1_*,2) < V(V(V(1,2),7),0) < V(V(V(1,3),0),1) < V(V(V(1,3),3),2) < V^{3}(1_*,3)\) and \(V^{3}(V^{2}(1_*,2+0)_*,21) = V(V(V(1,3),0),1)\) \(V^{3}(1_*,2+2) = V(V(V(1,3),3),2)\) Here are additional rules when using Index Recursion notation: Rule i5: \(nn = n\) for all \(n\) Rule i6: \(K^{c}(L(K^{d}(a_*,bm))_*,bn)\) implies * \(b\) is the value of the second parameter for all nested functions from \(K^{1}\) from \(K^{d - 1}\) * \(m\) is the value of the second parameter for nested function \(K^{d}\) * \(b\) is the value of the second parameter for all nested functions from \(K^{d + 1}\) from \(K^{c - 1}\) * \(n\) is the value of the second parameter for nested function \(K^{c}\) Rule i7: \(K^{c}(L(K^{d}(a+_*,b+m))_*,bn)\) where \(m < b + 1\) implies * \(b + 1\) is the value of the second parameter for all nested functions from \(K^{1}\) from \(K^{d - 1}\) * \(m\) is the value of the second parameter for nested function \(K^{d}\) * \(b\) is the value of the second parameter for all nested functions from \(K^{d + 1}\) from \(K^{c - 1}\) * \(n\) is the value of the second parameter for nested function \(K^{c}\) Correction (7 May 2019): * Note that \(b+n\) would be illegal. The \(+\) elaboration can only be used once and only by the innermost nested function. * Rule i7 will be reviewed. The latest Alpha Function spreadsheet does not follow this rule as defined here. Either the Alpha Function code will be changed or this rule will be updated. Correction (19 Aug 2019): * Added \(+\) elaboration to \(a+\). * Rule i7 has been corrected but Alpha Spreadsheet v4.0 code still needs changes to apply these rules. 'Summation Function' Summation Functions \(S(s)\) are useful short hand notation for summing a number of similar functions: I use them extensively for my Quantum Function and in particular the \(t()\) functions. Here are the basic rules: Rule s0: \(t_0(x,s)(t_0(x,0)) = 1\) for any \(x\) Rule s1: \(t_0(s)(t_0(1,0)) = t_0(1)\) Rule s2: \(t_0(s)(t_0(1,c + 1)) = t_0(1,c) + C = t_0(1,c) + t_0(s)(t_0(1,c))\) e.g. \(t_0(1,s)(t_0(2,c + 1)) = t_0(2,c) + C = t_0(2,c) + t_0(1,s)(t_0(2,c))\) Rule s3: \(t_0(c,s)(t_0(c + 1,0)) = t_0(c,0)\) e.g. \(t_0(1,s)(t_0(2,0)) = t_0(1,0)\) Rule s4: \(t_0(x,s)(t_0(x + 1,c + 1)) = t_0(x + 1,c) + C = t_0(x + 1,c) + t_0(x,s)(t_0(x + 1,c))\) Rule s4 is the general rule for Rule s2 where \(x = 0\) 'Further References' Further references to relevant blogs can be found here: User:B1mb0w Category:Blog posts